My interest is about polygons, where the number of vertices is a prime number, higher or equal to 5. When considering only star polygons, I wish to have a synthetic formula to compute the included circles radiuses, without even drawing any figure.
For a given prime number p, the number of related p-gons is given by the formula
\[ \Omega(p) = 1 + (p - 1) / 2 \]
Note that all those polygons are different. Among them, only the first one is a regular polygon. All others are star polygons, defined by the fact that their drawing pictures implies intersections between segments.
From the same set of p starting points on the unit circle, we can draw \(\Omega(p)\) p-gons, and each of them is unique. An easy way to draw them is to set a rule that allows to skip \(k\) points on each segment.
The regular polygon is of pace 1, and ties vertice #1 to vertice #2, and then vertice #2 to vertice #3 and so on.
The first star polygon is of pace 2, and ties vertice #1 to vertice #3, and then vertice #3 to vertice #5 and so on.
Each segment ties two and only two points of the unit circle. The pace has following properties
Let’s see an example, for a the case where \(p = 13\). Here, \(\Omega = 6\). So, there are one regular pentagon and five star pentagons. Here they are
Red dot on a regular polygon shows point numbered 1, that is the starting point we index other points from. As it exists a vertical axis symmetry, you may number clockwise or counter clockwise. On figures, I took the choice to number counter clockwise (mathematical way).
Two more complex examples. First, about 17-gons, second about 89-gons.